Bulletin of the Korean Mathematical Society (대한수학회보)

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ON 2-ABSORBING PRIMARY IDEALS IN COMMUTATIVE RINGS

  • Badawi, Ayman (Department of Mathematics & Statistics American University of Sharjah) ;
  • Tekir, Unsal (Department of Mathematics Marmara University) ;
  • Yetkin, Ece (Department of Mathematics Marmara University)
  • Received : 2013.09.23
  • Published : 2014.07.31
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Abstract

Let R be a commutative ring with $1{\neq}0$. In this paper, we introduce the concept of 2-absorbing primary ideal which is a generalization of primary ideal. A proper ideal I of R is called a 2-absorbing primary ideal of R if whenever $a,b,c{\in}R$ and $abc{\in}I$, then $ab{\in}I$ or $ac{\in}\sqrt{I}$ or $bc{\in}\sqrt{I}$. A number of results concerning 2-absorbing primary ideals and examples of 2-absorbing primary ideals are given.

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References

  1. D. D. Anderson and M. Bataineh, Generalizations of prime ideals, Comm. Algebra 36 (2008), no. 2, 686-696. https://doi.org/10.1080/00927870701724177
  2. D. F. Anderson and A. Badawi, On n-absorbing ideals of commutative rings, Comm. Algebra 39 (2011), no. 5, 1646-1672. https://doi.org/10.1080/00927871003738998
  3. A. Badawi, On 2-absorbing ideals of commutative rings, Bull. Austral. Math. Soc. 75 (2007), no. 3, 417-429. https://doi.org/10.1017/S0004972700039344
  4. A. Y. Darani and E. R. Puczylowski, On 2-absorbing commutative semigroups and their applications to rings, Semigroup Forum 86 (2013), no. 1, 83-91. https://doi.org/10.1007/s00233-012-9417-z
  5. M. Ebrahimpour and R. Nekooei, On generalizations of prime ideals, Comm. Algebra 40 (2012), no. 4, 1268-1279. https://doi.org/10.1080/00927872.2010.550794
  6. R. Gilmer, Multiplicative Ideal Theory, Queen Papers Pure Appl. Math. 90, Queen's University, Kingston, 1992.
  7. J. Huckaba, Rings with Zero-Divisors, New York/Basil, Marcel Dekker, 1988.
  8. S. Payrovi and S. Babaei, On the 2-absorbing ideals, Int. Math. Forum 7 (2012), no. 5-8, 265-271.

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